Asymptotic Problems with Boundary Effect in Kinetic Theory

动力学理论描述了大量粒子的动力学，比如空气粒子通过机翼的流动，或者核反应堆中的中子碰撞。以统计的方式，动力学描述连接了牛顿力学对粒子运动的微观尺度模拟和连续流体力学的宏观尺度模拟。本研究项目旨在开发新的数学方法来定量表征这些多尺度模型。研究的重点是稀薄气体、中子、电子和离子在周围环境的影响下在空间限定区域内的运动。它的应用范围从半导体或核聚变等高科技领域，到喷水器或荧光灯等日常生活设备。具体来说，这个项目集中于玻尔兹曼方程或输运方程的流体动力极限，即当一个小参数（Knudsen数或Strouhal数）趋近于零时，解是如何渐近变化的。在有界域中，动力学边界修正（即边界层）起着至关重要的作用。研究者打算证明在奇异边界层存在下渐近逼近的有效性。此外，还考虑了初始边界相互作用和非线性效应。为了解决非标准渐近展开，研究者寻求发展边界层几何校正的一般理论，重新缩放的剩余估计，米尔恩规则分析，非局部能量方法和边界层分解。这些技术可以进一步应用于Vlasov系统和磁流体动力学（MHD）方程。并给出了分数扩散和随机扩散的严格推导。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Kinetic theory describes the dynamics of a large number of particles, such as flows of air particles passing an airfoil, or neutrons' collisions in a nuclear reactor. In a statistical manner, the kinetic description bridges the micro-scale modeling of motion of particles by Newtonian mechanics and the macro-scale modeling by continuum fluid mechanics. This research project aims to develop novel mathematical methods to quantitatively characterize these multi-scale models. The investigation focuses on the motion of rarefied gas, neutrons, electrons, and ions, in spatially bounded regions under the influence of the surrounding environment. Its applications range from high-tech fields like semi-conductors or nuclear fusion, to daily-life devices like water sprays or fluorescent lamps.Specifically, this project concentrates on the hydrodynamic limit of the Boltzmann equation or transport equation, i.e. how the solution varies asymptotically when a small parameter, either the Knudsen number or the Strouhal number, approaches zero. In bounded domains, kinetic boundary corrections (i.e. boundary layers), play a crucial role. The investigator intends to justify the validity of the asymptotic approximation in the presence of singular boundary layers. Moreover, the initial-boundary interactions and nonlinear effects are taken into consideration. To tackle the non-standard asymptotic expansions, the investigator seeks to develop general theories of geometric correction in boundary layers, rescaled remainder estimates, Milne regularity analysis, non-local energy methods, and boundary layers decomposition. These techniques can be further applied to Vlasov systems and magnetohydrodynamical (MHD) equations. Also, the rigorous derivations of fractional diffusion and stochastic diffusion are involved.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.